626 ON THE PROBLEM OF RANDOM VIBRATIONS, ETC. [441
and thus
(r*) r***® l - 2
As before, the leading term on the right is
J — _ ._ g-JrrySfs) .......^.................../Ygx
and the other integrals can be derived from it by differentiations with respect to 2 (s). So far as the first two terms inclusive, we find
from which we may fall back upon (45) by dropping the S and making p — n. In general X (p) — n. The approximation could be pursued.
Let us now suppose that the representative points are distributed over the area of a circle of radius L, all infinitesimal equal areas being equally probable. Of the total n the number (p) which fall between I and .1 + dl should be n . (ZldlfL2), and thus
Introducing these values in (74), we get
A similar extension may be made in the problem where the component vectors are drawn in three dimensions.
* The applicability of the second term {in l/?i) to the case of an entirely random distribution over the area of the circle "L is not over secure.ds finite, thouh small,